The inverse scattering problems aim to determine the unknown objects from the measured wave fields. In many cases of practical interest, it is difficult or even impossible to obtain the full-aperture measurements all around the unknown objects, this motivates us to consider the so-called limited-aperture problems. The  limited-aperture problems are fundamental in applications such as radar, sonar, geophysical exploration, medical imaging and nondestructive testing. Various reconstruction algorithms have been developed
[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]} using the limited-aperture data directly.
Although uniqueness of the inverse scattering problems can be proved in some cases [13]}, the performance of the reconstruction algorithms is not entirely satisfactory.
A typical feature is that the “shadow region” is elongated in down range [4]}, [7]}.
Physically, the information from the “shadow region” is very weak, especially for high frequency waves [9]}.
For the two-dimensional problems, the numerical
experiments of the decomposition methods in [11]}, [12]} indicate that satisfactory reconstructions need an aperture not smaller than 180 degrees.
